The key modification compared to [1]} is that the
step length is not uniform but it is capped so that we do not exit \(K\) .
As we walk orthogonal to the current point, one has \(\Vert \mathbf {x}_{t+1}\Vert _2^2 = \Vert \mathbf {x}_t\Vert _2^2 + \delta _t^2\)  for all \(t\) 
and so \(\sum _t \delta _t^2 \le 4n\) . Let us call a step \(t\)  long if \(\delta _t = 1\) 
and short otherwise. So we cannot have more than \(4n\)  many long steps. But in each short step (i.e. \(\delta _r = r_t\) )
we have a probability of at least 1/2 that some coordinate or constraint becomes tight
and this cannot happen more than \(\frac{7}{8}n\)  times before exiting at (2).
So the probability of reaching \((8)\)  is less than \((1/2)^n\) .
